What diameter should a 10-m-long steel wire have if we do not want it to stretch more than 0.5 cm under a tension of 940 N? Take Young's modulus of steel as 20 1010 Pa.
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Answer:
What diameter should a 10-m-long steel wire have if we do not want it to stretch more than 0.5 cm under these conditions? The cable has Young's modulus of 20 × 1010 N/m2. Numerically we find d = 3.56 mm (or equivalently r = d/2=1.77 mm).
Explanation:
We have the maximum velocity given by vmax = ωA. Thus vmax = 2π × 5 cm/s, which
numerically gives 31.4 cm/s
The time taken by the particle to reach its equilibrium is one-quarter the period. To prove this
(although you don’t have to) consider the equation of motion: x = A cos(wt + φ), where in this
case obviously we have φ = 0 (initial conditions) so that x(0) = A.
Solving for x = 0 we have cos ωt = 0, hence we easily find ωt = π/2 (first occurrence), which
means t = π/(2ω) = T/4.
Numerically t = π/(2 × 2π)s = 1/4s = 0.25s
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